
where (REF ) follows by Corollary REF  and
Theorem 8.10 [Slutsky's Theorem] (pg. 58) of [1]}.
Now consider all \(x \in [\Delta +1, \ldots , L]\)  to write,
\(\sqrt{n} \begin{bmatrix} \lbrace  \ln S_n(\Delta + 1) - \ln S(\Delta + 1) \rbrace  \\\vdots \\\lbrace  \ln S_n(L) - \ln S(L) \rbrace \end{bmatrix} = \mathbf {K} \times \sqrt{n} (\hat{\mathbf {\Lambda }}_n -\mathbf {\Lambda }) + o_p(1).\) 
